Question: $\dfrac{d}{dx}\left(e^x\cos(x)\right)=$
Explanation: $e^x\cos(x)$ is the product of two, more basic, expressions: $e^x$ and $\cos(x)$. Therefore, the derivative of the expression can be found using the product rule : $\begin{aligned} \dfrac{d}{dx}[u(x)v(x)]&=\dfrac{d}{dx}[u(x)]v(x)+u(x)\dfrac{d}{dx}[v(x)] \\\\ &=u'(x)v(x)+u(x)v'(x) \end{aligned}$ $\begin{aligned} &\phantom{=}\dfrac{d}{dx}\left(e^x\cos(x)\right) \\\\ &=\dfrac{d}{dx}\left(e^x\right)\cos(x)+e^x\dfrac{d}{dx}(\cos(x))&&\gray{\text{The product rule}} \\\\ &=e^x\cdot\cos(x)+e^x\cdot (-\sin(x))&&\gray{\text{Differentiate }e^x\text{ and }\cos(x)} \\\\ &=e^x(\cos(x)-\sin(x))&&\gray{\text{Simplify}} \end{aligned}$ In conclusion, $\dfrac{d}{dx}\left(e^x\cos(x)\right)=e^x(\cos(x)-\sin(x))$ or any other equivalent form.